Bistable travelling waves for nonlocal reaction diffusion equations

We are concerned with travelling wave solutions arising in a reaction diffusion equation with bistable and nonlocal nonlinearity, for which the comparison principle does not hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We construct a travelling wave solution connecting 0 to an unknown steady state, which is"above and away", from the intermediate equilibrium. For focusing kernels we prove that, as expected, the wave connects 0 to 1. Our results also apply readily to the nonlocal ignition case.


Introduction
We consider the nonlocal bistable reaction diffusion equation where 0 < θ < 1. Here φ * u(x) := R u(x − y)φ(y) dy, with φ a given bounded kernel such that We are looking for travelling waves solutions supported by the integro-differential equation (1), that is a speed c * ∈ R and a smooth U such that supplemented with the boundary conditions In this work we construct such a travelling wave solution, and show that the behavior on the left is improved to lim x→−∞ U (x) = 1 for focusing kernels. Our results also apply readily to the nonlocal ignition case (see below).

Equation (1) is a nonlocal version of the well known reaction-diffusion equation
with the bistable nonlinearity f (s) := s(s − θ)(1 − s). Homogeneous reaction diffusion equations have been extensively studied in the literature (see [14], [3,4], [10], [6], [18] among others) and are known to support the existence of monotone travelling fronts for three classes of nonlinearity: bistable, ignition and monostable. Moreover, for bistable and ignition nonlinearities, there exists a unique front speed c * whereas, for monostable nonlinearities, there exists a critical speed c * such that all speeds c ≥ c * are admissible. In this local context, many techniques based on the comparison principle -such as some monotone iterative schemes or the sliding method [7]-can be used to get a priori bounds, existence and monotonicity properties of the fronts. Recently, much attention was devoted to the introduction of a nonlocal effect into the nonlinear reaction term. From the mathematical point of view, the analysis is quite involved since integrodifferential equations with a nonlocal competition term generally do not satisfy the comparison principle. In [5], Berestycki, Nadin, Perthame and Ryzhik have considered the following non-local version of the Fisher-KPP equation They prove that equation (6) admits a critical speed c * so that, for all c ≥ c * , there exists a travelling wave (c, U ) solution of In favorable situations, namely when the steady state u ≡ 1 remains linearly stable, they further obtain lim x→−∞ U (x) = 1. Nevertheless, the positive steady state u ≡ 1 may present, for some kernels, a Turing instability (see e.g. [11], [5], [1]). In such situations, it was proved in [9] and in [1] that the waves with large speeds actually connect the two unstable states 0 and 1. Notice that the former work considers kernels with exponential decay and uses monotonicity arguments inspired by [13], whereas the latter uses more direct arguments which allow kernels with algebraic decay. Concerning this issue of the behavior of the wave on the left, we also refer the reader to [17], [16]. In a related framework, the authors of the present work have constructed curved fronts for nonlocal reaction diffusion equations [2] of the form In population dynamics, such equations serve as prototypes of models for structured populations evolving in a environmental cline.
In view of the existence of fronts for both the nonlocal Fisher-KPP equation (6) and the local bistable equation (5), it is then expected that the nonlocal bistable equation (1) supports the existence of travelling waves. In this work, we shall construct such a solution. It is worth being mentioned that, among other things, nonlinearities such as u(φ * u − θ)(1 − u) are treated in [19]. Notice that our equation does not fall into [19, equation (1.6) [19, hypothesis (H1)], which actually provides the stability of both u ≡ 0 and ≡ 1.
Let us now state our main result on the existence of a travelling wave solution.
Theorem 1.1 (A bistable travelling wave) There exist a speed c * ∈ R and a positive profile such that, for some ε > 0, U is decreasing on [x, +∞) for somex > 0, and Now, if the kernel tends to the Dirac mass, we expect the above travelling wave to be a perturbation of the underlying wave for the local case, namely (c * 0 , U 0 ) the unique solution of and so to satisfy lim x→−∞ U (x) = 1. Our next result states such a behavior assuming c * 0 = 0, which is equivalent to θ = 1 2 . To make this perturbation analysis precise, we take σ > 0 as a focusing parameter, define and are interested in the asymptotics σ → 0.
Proposition 1.2 (Focusing kernels) Denote by (c * σ , U σ ) the travelling wave associated with the kernel φ σ , as constructed in Theorem 1.1.
(ii) Assume θ = 1 2 and R z 2 φ(z) dz < ∞. Then there is σ 0 > 0 such that, for all 0 < σ < σ 0 , Remark 1.3 (Ignition case) While proving the above results for the bistable case, it will become clear that the same (with the additional information c * > 0) holds for the ignition case, that is for which proofs are simpler. This will be clarified in Section 6.
Let us comment on the main result, Theorem 1.1. Due to the lack of comparison principle, the construction of a travelling wave solution is based on a topological degree argument, a method introduced initially in [6]. After establishing a series of a priori estimates, it enables to construct a solution in a bounded box. Then we let the size of the box tend to infinity to construct a solution on the whole line. In contrast with [5] and because of the intermediate equilibrium u ≡ θ, it is far from clear that the constructed wave is non trivial -or, equivalently, that it "visits" both (0, θ) and (θ, 1). Such an additional difficulty also arises in the construction of bistable waves in cylinders [8], where the authors use energy arguments to exclude the possibility of triviality. This seems not to be applicable to our nonlocal case. Our arguments are rather direct and are based on the sharp property of Proposition 3.1 and the construction of bump-like sub-solutions in Lemma 3.2.
The organization of the paper is as follows. In Section 2, we construct a solution u on a bounded interval thanks to a Leray-Schauder topological degree argument. In Section 3, we show that, when we let the bounded interval tend to the whole line, the limit profile U is non trivial. The behaviors (8) and (9) are then proved in Section 4. We investigate the case of the focusing kernels, that is Proposition 1.2, in Section 5. Last, in Section 6 we indicate how to handle the ignition case.

Construction of a solution u in a box
Notice that the methods used in this section are inspired by [5].
For a > 0 and 0 ≤ τ ≤ 1, we consider the problem of finding a speed c = c a τ ∈ R and a profile u = u a τ : [−a, a] → R such that whereū denotes the extension of u equal to 1 on (−∞, −a) and 0 on (a, ∞) (in the sequel, for ease of notation, we always write u in place ofū). This realizes a homotopy from a local problem (τ = 0) to our nonlocal problem (τ = 1) in the box (−a, a). We shall construct a solution to P 1 (a) by using a Leray-Schauder topological degree argument. If (c, u) is a solution achieving a negative minimum at x m then x m ∈ (−a, a) and −u ′′ − cu ′ = 0 on a neighborhood of x m . The maximum principle thus implies u ≡ u(x m ), which cannot be. Therefore any solution of P τ (a) satisfies u ≥ 0 and, by the strong maximum principle,

A priori bounds of solutions in the box
The following lemma provides a priori bounds for u. Proof. If τ = 0 we directly get 0 ≤ u ≤ 1 for the local problem. Now, for 0 < τ ≤ 1, assume M := max x∈[−a,a] u(x) > 1 (otherwise there is nothing to prove). In view of the boundary conditions, there is a x m ∈ (−a, a) such that M = u(x m ). Evaluating (12) at x m we see that φ * u(x m ) ≤ 1. Now since u ≥ 0, we also have Let us first assume that Using u ′ (x m ) = 0, isolating u ′ (x) and integrating again from x to x m , we discover Using u(x m ) = M and isolating u(x), we get after elementary computations where B(y) := e −y +y−1 Observe that B(y) ≤ 1 2 for y > 0 so that and in particular, Now we define where we have used (14). From the definition of x 0 we deduce that This concludes the proof in the case c < 0. The case c > 0 can be treated in a similar way by we directly obtain (14) and we can repeat the above arguments. This completes the proof of the lemma.
We now provide a priori bounds for the speed c.
We now provide a priori bounds for the speed c. We will prove two separate estimates. Proof. Let a > 0 be given. We consider a solution (c, u) of P τ (a). It satisfies: We see that v(0) → 1 as c → −∞. It follows that, for any a > 0, there existsc min (a) > 0 such that c ≤ −c min (a) implies θ < v(0) ≤ u(0), so that u cannot solve P τ (a). Hence, all solutions (c, u) of P τ (a) with 0 ≤ τ ≤ 1 are such that c ≥ −c min (a). Proof. We assume that c ≤ −1 (otherwise there is nothing to prove), and define M > 0 and a 0 > 0 as in Lemma 2.1. The first step of the proof is to find uniform bounds on u ′ , following ideas from [5, Lemma 3.3]. We first notice that (e cx u ′ (x)) ′ = e cx (u ′′ (x) + cu ′ (x)), and an integration of this expression provides for x > y: where we have chosen x = a, and used the fact that u ′ (a) ≤ 0 to obtain this last estimate. Next, define Indeed, assume by contradiction that there are some Integrating this from y to a and using u(a) = 0 we see that which contradicts the definition of K 0 . This proves (20).
If on the contrary, u is non-increasing on R, which, combined to (20) and (19) implies so that in any case, c min := − 128QR θ(1−θ) 2 is an explicit lower bound for c.

Construction of a solution in the box
Equipped with the above a priori estimates, we now use a Leray-Schauder topological degree argument (see e.g. [6], [5] or [2] for related arguments) to construct a solution (c, u) to P 1 (a).

Proposition 2.5 (A solution in the box)
There exist K > 0 and a 0 > 0 such that, for all a ≥ a 0 , Problem P 1 (a) admits a solution (c, u), that is which is such that Proof. For a given nonnegative function v defined on (−a, a) and satisfying the Dirichlet boundary conditions as requested in P τ (a) -that is v(−a) = 1 and v(a) = 0-consider the family 0 ≤ τ ≤ 1 of linear problems Denote by K τ the mapping of the Banach space X : Constructing a solution (c, u) of P 1 (a) is equivalent to showing that the kernel of Id − K 1 is nontrivial. The operator K τ is compact and depends continuously on the parameter 0 ≤ τ ≤ 1.
Thus the Leray-Schauder topological argument can be applied. Define the open set It follows from the a priori estimates Lemma 2.1, Lemma 2.2 and Lemma 2.3, that there exists a 0 > 0 such that, for any a ≥ a 0 , any 0 ≤ τ ≤ 1, the operator Id − K τ cannot vanish on the boundary ∂S. By the homotopy invariance of the degree we thus have deg(Id − K 1 , S, 0) = deg(Id − K 0 , S, 0).
To conclude, observe that we can compute and that u c 0 (0) is decreasing with respect to c (in particular there is a unique c 0 such that u c0 0 (0) = θ). Hence by using two additional homotopies (see [5] or [2] for details) we can compute deg(Id − K 0 , S, 0) = −1 so that deg(Id − K 1 , S, 0) = −1 and there is a (c, u) ∈ S solution of P 1 (a). Finally, Lemma 2.4 provides a lower bound c ≥ −c min , uniform in a ≥ a 0 . This concludes the proof of the proposition.
A solution on R. Equipped with the solution (c, u) of P 1 (a) of Proposition 2.5, we now let a → +∞. This enables to construct -passing to a subsequence a n → +∞-a speed −c min ≤ c * ≤ c max and a function U : In contrast with the nonlocal Fisher-KPP equation considered in [5] we need additional arguments to show that the constructed U is non trivial, i.e. that U "visits" both (0, θ) and (θ, 1). This is the purpose of the next section.

Non triviality of U the solution on R
In this section, we provide additional a priori estimates on the solution u in the box (−a, a), which in turn will imply the non triviality of the solution U on R.
First, using the homotopy of the previous section, we show that the solution in the box cannot attain θ elsewhere that at x = 0. Proof. From Proposition 2.5, we know that there is a solution (c τ , u τ ) of and that (c τ , u τ ) depends continuously upon 0 ≤ τ ≤ 1. For τ = 0, in view of (25), the solution u 0 satisfies u 0 (x) = θ if and only if x = 0. We can therefore define Assume by contradiction that there is a x * = 0 such that u τ * (x * ) = θ. Without loss of generality, we can assume x * < 0 and u τ * > θ on (x * , 0). By the definition of τ * as a supremum, one must have u τ * ≥ θ on (−a, 0), which in turn enforces u ′ τ * (x * ) = 0. Hence v := u τ * − θ is positive on (x * , 0), zero at x * and satisfies the linear elliptic equation It then follows -see e.g. [12,Lemma 3.4]-that v ′ (x * ) > 0, which is a contradiction. Hence u τ * attains θ only at x = 0. To conclude let us prove that τ * = 1.
We now construct a subsolution of a linear equation, having the form of a bump, that will be very useful in the following.
and starts to decreases afterx. On [0,x], we have tan Hence ψ/ψ(x) is as requested.
We will also use the elementary following lemma.
Proof. One solves the linear ODE and sees that the function x is as requested.
We now show that u can be uniformly (with respect to a) bounded away from θ far on the right or the left, depending on the sign of the speed c.
Assume c > −2 √ κ and let us prove (i). Since φ ∈ L 1 , there exists R > 0 such that , where M is the L ∞ bound we have on u. We turn upside down the bump and make it slide from the right towards the left until touching the solution u. Precisely one can define We aim at proving that α 0 ≤ R uniformly with respect to large a. Assume by contradiction that α 0 > R. The function v := u − θ + θ 2 ψ(· − α 0 ) has a zero maximum at some point x 0 . Notice that since ψ(x) = max ψ, the definition of α 0 implies that x 0 − α 0 ∈ (0,x], so that ψ is a subsolution of (29) around Now, since α 0 ≥ R, we have x 0 ≥ R and we can estimate the nonlocal term by Since θ 2 ≤ u(x 0 ) < θ, it follows from (32) that 0 ≥ θ 2 1−θ 2 − κ, which contradicts the definition of κ. As a result α 0 ≤ R, which means that the minimum θ/2 of the reversed bump can slide to the left at least until R +x. In other words we have u ≤ θ/2 on [R +x, a] which concludes the proof of (i).
Assume c < 2 √ κ and let us prove (ii). Since φ ∈ L 1 , we can choose R > 0 such that where M is the L ∞ bound we have on u. Before using the bump we need a preliminary result via the function χ of Lemma 3.3.
Non triviality of U . Let us recall that (c * , U ) is constructed as the limit of (c a , u a ) as a → ∞. By extraction if necessary we can assume that the (c a , u a )' satisfy either (i) or (ii) of Proposition 3.4, and so does (c * , U ). As a result, the constructed wave (c * , U ) is non trivial.

Behaviors of U in ±∞
We now prove the behavior (8) as x → −∞, the limit (9) as x → ∞, and that the constructed front is decreasing for x > 0 large enough. This will complete the proof of Theorem 1.1. Moreover, there existsx > 0 such that U is decreasing on [x, ∞).
Consider first the case where c * ≥ 0. Since U is a limit of solutions of Proposition 2.5, the Proposition 3.4 shows that there exists ε > 0 such that U ≤ θ/2 on (1/ε, ∞). To investigate the left side, as in the proof of Proposition 3.4 (ii), we choose R > 0 such that [−R,R] c φ ≤ 1−θ 4M , where M is the L ∞ bound we have on U . By the Harnack inequality applied to U − θ, there exists C > 0 such that, for all x 0 ≤ −2R, Let us now define, for η ≥ 0, Since ψ η ≤ U on (−∞, −2R] for η > 0 large enough, we can define Let us assume by contradiction that η 0 > 0. The function U − ψ η0 then attains a zero minimum at a point x 0 < −2R (notice that γ ≤ γ 1 prevents x 0 = −2R). Hence where we have used the estimate (36) for U (notice that this is possible since we have the two ingredients (37) and U (x 0 ) = ψ η0 (x 0 ) ≤ θ + 1−θ 4C ). This is a contradiction which proves that η 0 = 0, and then U ≥ θ + γ on (−∞, −2R). This concludes the case c * ≥ 0.
Consider next the case where c * ≤ 0. Since U is a limit of solutions of Proposition 2.5, the Proposition 3.4 shows that there exists ε > 0 such that U ≥ θ + ε on (−∞, 1/ε). To investigate the right side, as in the proof of Proposition 3.4 (i), we choose R > 0 such that which satisfies ψ η ≥ U on [2R, ∞) for η > 0 large enough. We can then define Let us assume by contradiction that η 0 > 0. The function U − ψ η0 then attains a zero maximum at a point x 0 > 2R, and therefore where we have used the estimate (33) for U . This is a contradiction which proves that η 0 = 0, and then U ≤ θ − γ on (2R, ∞). This concludes the case c * ≤ 0.
Step 3: We show that U decreases to 0 on some interval (x, ∞). Choose R > 0 large enough so that . Assume by contradiction that U admits a local maximum at some point x m ≥ R. Since U (x m ) < θ, by evaluating the equation (7) we see that 1 ≤ φ * U (x m ). But on the other hand which is a contradiction. Hence U cannot attain a maximum on (R, ∞), which in turn implies that there isx > 0 such that U is monotonic (increasing or decreasing) on [x, ∞). Hence, as x → ∞, U (x) → l and, by Step 2, 0 ≤ l ≤ θ − ε.
We define next v n (x) := U (x + n), which solves Since the L ∞ norm of the right hand side member is uniformly bounded with respect to n, the interior elliptic estimates imply that, for all R > 0, all 1 < p < ∞, the sequence (v n ) is bounded in . From Sobolev embedding theorem, one can extract v ϕ(n) → v strongly in C 1,β loc (R) and weakly in W 2,p loc (R). Since v n (x) = U (x + n) → l we have v ≡ l and v ′ ≡ 0. Combining this with the fact that v solves we have l(l − θ)(1 − l) = 0, which implies l = 0 and the decrease of U on [x, ∞).

Focusing kernels
In this section we consider (c * σ , U σ ) the constructed waves for the focusing kernels We prove Proposition 1.2. Item (i) consists in a perturbation analysis, and item (ii) will follow from the L 2 analysis performed in [1].
Lemma 5.1 (Uniform bounds for (c * σ , U σ )) Let σ 0 > 0 be arbitrary. Then there is M > 0, c min ∈ R, c max ∈ R such that, for all σ ∈ (0, σ 0 ), Proof. It is sufficient to work on the solutions (c σ , u σ ) in the box. Define M σ := max x∈[−a,a] u σ (x). A first lecture of Lemma 2.1 yields the rough bound (17) with the kernel φ σ in place of φ. (17) that there is a constant b > 0, such that M σ ≤ b 2 /σ 2 . Equipped with this rough bound, we go back to the proof of Lemma 2.1 but rather than (16) we select the improvement Hence, going further into the proof, we discover , that is a uniform bound M for M σ as σ → 0. In view of Lemma 2.2 and of the proof of Lemma 2.4, the uniform bound M yields uniform bounds c max and c min for the speed c σ . The lemma is proved.

The ignition case
Here we explain briefly how to use similar arguments to handle the case of the ignition case.
To conclude, the behavior (8) as x → −∞ is proved as in Proposition 4.1.